We define the class of fully $\omega_1$-$p^{\omega+n}$-projective abelian $p$-groups and establish its crucial properties. It is shown that this class is situated between the classes of strongly $\omega_1$-$p^{\omega+n}$-projective and $\omega_1$-$p^{\omega+n}$-projective abelian $p$-groups, and it was constructed a fully $\omega_1$-$p^{\omega+n}$-projective group that is not strongly $\omega_1$-$p^{\omega+n}$-projective thus showing that one of the inclusions is proper. These results strengthen theorems due to Keef in J. Algebra Numb. Theory Acad. (2010) and also continue our own achievements in Hacettepe J. Math. Stat. (2014).